Abstract

Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and K a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I - T) is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann's algorithm for T under suitable control conditions. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.

2000 AMS Subject Classification: 47H09; 47H10.

Keywords

1 Introduction

Let E and E* be a real Banach space and the dual space of E, respectively. Let K be a nonempty subset of E. Let J denote the normalized duality mapping from E into 2E* given by J(x) = {f∈E* : 〈x, f〉 = ||x||2 = ||f||2}, for all x∈E, where 〈·,·〉 denotes the duality pairing between E and E*. In the sequel, we will denote the set of fixed points of a mapping T : K → K by F (T) = {x∈K : Tx = x}.

A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence {κn}n=1∞⊆[1,∞) such that limn→∞κn = 1 (see, e.g., [1–3]) if for all x, y∈K, there exist a constant κ∈ [0, 1) and j(x - y) ∈J(x - y) such that

⟨Tnx-Tny,j(x-y)⟩≤κn∥x-y∥2-κ∥x-y-(Tnx-Tny)∥2,∀n≥1.

(1)

If I denotes the identity operator, then (1) can be written as

⟨(I-Tn)x-(I-Tn)y,j(x-y)⟩≥κ∥(I-Tn)x-(I-Tn)y∥2-(κn-1)∥x-y∥2,∀n≥1.

(2)

The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j is the identity and it is shown by Osilike et al. [2] that (1) (and hence (2)) is equivalent to the inequality

∥Tnx-Tny∥2≤λn∥x-y∥2+λ∥x-y-(Tnx-Tny)∥2,

where limn→∞λn = limn→∞[1 + 2(κn- 1)] = 1, λ = (1 - 2κ) ∈ [0, 1).

A mapping T with domain D(T) and range R(T) in E is called strictly pseudocontractive of Browder-Petryshyn type [4], if for all x, y∈D(T), there exists κ∈ [0, 1) and j(x - y) ∈J(x - y) such that

⟨Tx-Ty,j(x-y)⟩≤∥x-y∥2-κ∥x-y-(Tx-Ty)∥2.

(3)

If I denotes the identity operator, then (3) can be written as

⟨(I-T)x-(I-T)y,j(x-y)⟩≥κ∥(I-T)x-(I-T)y∥2.

(4)

In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality

∥Tx-Ty∥2≤∥x-y∥2+k∥x-y-(Tx-Ty)∥2,k=(1-2κ)<1,

It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.

A mapping T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that

∥Tnx-Tny∥≤L∥x-y∥,n≥1

for all x, y∈K and is said to be demiclosed at a point p if whenever {xn} ⊂D(T) such that {xn} converges weakly to x∈D(T ) and {Txn} converges strongly to p, then Tx = p.

Kim and Xu [6] studied weak and strong convergence theorems for the class of asymptotically κ-strictly pseudocontractive mappings in Hilbert space. They obtained a weak convergence theorem of modified Mann iterative processes for this class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection method. They proved the following.

Assume that the control sequence {αn}n=1∞ is chosen in such a way that κ + λ ≤ αn≤ 1 -λ for all n, where λ∈ (0, 1) is a small enough constant. Then, {xn} converges weakly to a fixed point of T.

The modified Mann's iteration scheme was introduced by Schu [7, 8] and has been used by several authors (see, for example, [1–3, 9–11]). One question is raised naturally: is the result in Theorem KX true in the framework of the much general Banach space?

Osilike et al. [5] proved the convergence theorems of modified Mann iteration method in the framework of q-uniformly smooth Banach spaces which are also uniformly convex. They also obtained that a modified Mann iterative process {xn} converges weakly to a fixed point of T under suitable control conditions. However, the control sequence {αn} ⊂ [0,1] depended on the Lipschizian constant L and excluded the natural choice αn=1n,n≥1. These are motivations for us to improve the results. We prove the demiclosedness principle and weak convergence theorem of the modified Mann's algorithm for T in the framework of uniformly convex Banach spaces which have the Fréchet differentiable norm. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.

We will use the notation:

1.

⇀ for weak convergence.

2.

ωW(xn)={x:∃xnj⇀x} denotes the weak ω-limit set of {xn}.

2 Preliminaries

Let E be a real Banach space. The space E is called uniformly convex if for each ε > 0, there exists a δ > 0 such that for x, y∈E with ||x|| ≤ 1, ||y|| ≤ 1, ||x- y|| ≥ ε, we have ∥12(x+y)∥≤1-δ. The modulus of convexity of E is defined by

δE(ε)=inf{1-∥12(x+y)∥:∥x∥≤1,∥y∥≤1,∥x-y∥≥ε,}∀x,y∈E

for all ε∈ [0,2]. E is uniformly convex if δE(0) = 0 and δE(ε) > 0 for all ε∈ (0, 2]. The modulus of smoothness of E is the function ρE : [0, ∞) ∈ [0, ∞) defined by

ρE(τ)=sup{12(∥x+y∥+∥x-y∥)-1:∥x∥≤1,∥y∥≤τ},∀x,y∈E.

E is uniformly smooth if and only if limτ→0ρE(τ)τ=0.

E is said to have a Fréchet differentiable norm if for all x∈U = {x∈E : ||x|| = 1}

limt→0∥x+ty∥-∥x∥t

exists and is attained uniformly in y∈U. In this case, there exists an increasing function b : [0, ∞) → [0, ∞) with limt→0[b(t)∕t]=0 such that for all x, h∈E

12∥x∥2+⟨h,j(x)⟩≤12∥x+h∥2≤12∥x∥2+⟨h,j(x)⟩+b(∥h∥).

(5)

It is well known (see, for example, [[12], p. 107]) that uniformly smooth Banach space has a Fréchet differentiable norm.

Lemma 2.3 [[13], p. 29] Let K be a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space E, and let T : K → E be a nonexpansive mappings. Let {xn} be a sequence in K such that {xn} converges weakly to some point x∈K. Then, there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that

where τn=[1+2α(κn-1)]12. (In fact, in (7) the domain of β*(·) requires ||x - y - (Tnx-Tny)|| ≠ 0. But when ||x - y - (Tnx-Tny)|| = 0, we have ||Tα,nx-Tα,ny||2 = ||x - y||2, which still satisfies the inequality ∥Tα,nx-Tα,ny∥2≤τn2∥x-y∥2. So we do not specially emphasize the situation that the argument of β*(·) equals 0 in this inequality and the following proof of Theorem 3.1.) Define Gα,m : K → E by

Gα,mx=1τmTα,mx,m≥1.

Then, Gα,m is nonexpansive and it follows from Lemma 2.3 that there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that

Since T is continuous, we have (I - T)(p) = 0, completing the proof of Lemma 2.6. □

Lemma 2.7 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping with F(T) ≠ ∅. Let {xn}n=1∞ be the sequence satisfying the following conditions:

(a)

limn→∞∥xn-p∥exists for every p∈F(T );

(b)

limn→∞∥xn-Txn∥=0;

(c)

limn→∞∥txn+(1-t)p1-p2∥exists for all t∈ [0, 1] and for all p1, p2∈F (T ).

Then, the sequence {xn} converges weakly to a fixed point of T.

Proof. Since limn→∞ ||xn - p|| exists, then {xn} is bounded. By (b) and Lemma 2.6, we have ωW(xn)⊂F(T). Assume that p1,p2∈ωW(xn) and that {xni} and {xmj} are subsequences of {xn} such that xni⇀p1 and xmj⇀p2, respectively. Since E has the Fréchet differentiable norm, by setting x = p1 - p2, h = t(xn - p1) in (5) we obtain

3 Main results

Theorem 3.1 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence {κn}n=1∞⊂[1,∞), such that ∑n=1∞(κn-1)<∞, and let F(T) ≠ ∅. Assume that the control sequence {αn}n=1∞ is chosen so that

(i*)

0 < αn< κ, n ≥ 1;

(ii*)

∑n=1∞αn(κ-αn)=∞. (11)

Given x1∈K, then the sequence {xn}n=1∞ is generated by the modified Mann's algorithm:

xn+1=(1-αn)xn+αnTnxn,

(12)

converges weakly to a fixed point of T.

Proof. Pick a p∈F(T). We firstly show that limn→∞ ||xn - p|| exists. To see this, using (2) and (6), we obtain

Since E is uniformly convex, then δ(s)s is nondecreasing, and since (∏j=nn+m-1λj)∥xn-p1∥≤(∏j=nn+m-1λj)λn-1∥xn-1-p1∥≤⋅⋅⋅≤(∏j=nn+m-1λj)(∏j=1n-1λj)∥x1-p1∥=(∏j=1n+m-1λj)∥x1-p1∥, hence it follows from (18) that

Now, apply Lemma 2.7 to conclude that {xn} converges weakly to a fixed point of T. □

Theorem 3.2 Let E be a real Banach space with the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence {κn} ⊂ [1, ∞) such that ∑n=1∞(κn-1)<∞, let F(T) ≠ ∅. Let {αn} be a real sequence satisfying the condition (11). Given x1∈K, let {xn}n=1∞ be the sequence generated by the modified Mann's algorithm (12). Then, the sequence {xn} converges strongly to a fixed point of T if and only if

liminfn→∞d(xn,F(T))=0,

where d(xn, F(T)) = infp∈F(T)||xn - p||.

Proof. In the real Banach space E with the Fréchet differentiable norm, we still have

Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.